Mersenne number

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

Mersenne prime

A prime number of the form $M_n=2^n-1$, where $n=1,2,\ldots$. Mersenne numbers were considered in the 17th century by M. Mersenne. The numbers $M_n$ can be prime only for prime values of $n$. For $n=2,3,5,7$ one obtains the prime numbers $M_n=3,7,31,127$. However, for $n=11$ the number $M_n$ is composite. For prime values of $n$ larger than $11$, among the $M_n$ one encounters both prime and composite numbers. The fast growth of the numbers $M_n$ makes their study difficult. By considering concrete numbers $M_n$ it has been shown, for example, that $M_{31}$ (L. Euler, 1750) and $M_{61}$ (I.M. Pervushin, 1883) are Mersenne numbers. Computers were used to find other very large Mersenne numbers, among them $M_{11213}$. The existence of an infinite set of Mersenne numbers is still an open problem (1989). This problem is closely related with the problem on the existence of perfect numbers.

References

[1]	H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)
[2]	A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)

Comments

Presently (1989) it is known that for the following $n$ the Mersenne number $M_n$ is prime: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 132049, 216091. See [a1].

The Lucas test provides a very simple method to establish primality of these numbers. This test consists of the following (cf. [a2]). Define $S_1=4$ and $S_{k+1}=S_k^2-2$ for $k\geq1$. Then $M_n$ is prime if and only if $M_n$ divides $S_{n-1}$ (and $n$ is a prime number).

References

[a1]	H. Riesel, "Prime numbers and computer methods for factorisation" , Birkhäuser (1986)
[a2]	D. Shanks, "Solved and unsolved problems in number theory" , Chelsea, reprint (1978)